Linear instability of periodic orbits of free period Lagrangian systems

1.   Introduction and description of the problem

A celebrated result proved by Poincaré at the beginning of the last century puts on evidence the relation between the (linear and exponential) instability of an orientation preserving closed geodesic as a critical point of the geodesic energy functional on the free loop space of a surface and the minimization property of such a critical point. The literature on this criterion is quite broad. We refer the interested reader to ^[1,2,3] and references therein.

A quite natural question is related to understand the role played by the energy level $h$ on the instability properties of a periodic orbit of an autonomous Lagrangian system. Some years ago, in his expository article ^[4], Abbondandolo studied the question of the existence of periodic orbits of the free period Lagrangian systems in terms of the Mañé critical values. It is worth noticing that the fixed energy problem for Tonelli Lagrangians as well as for the restricted class of magnetic systems has been intensively studied by many outstanding mathematicians such as Arnold, Novikov, Ginzburg, Taimanov, Contreras, Paternain, etc. by starting on the work of Arnol'd ^[5] in the 1960s. If on the one hand it is impossible to keep track of all the literature on the topic, some breakthrough papers in the field are ^[6] and ^[7].

Very recently, Ureña in ^[8], studied the instability of closed orbits obtained by minimization, for an autonomous Lagrangian system by combining the classical principle of Jacobi-Maupertuis principle together with a nice reduction argument firstly introduced by Carathéodory. In this way, under some suitable conditions, the dynamics of a free period Lagrangian system can be seen as the dynamics of a non-autonomous and fixed period Lagrangian system lowering by $1$ the degrees of freedom.

1.1.   Basic definitions and main result

Let $(M, \langle{{\cdot}}, {{\cdot}} \rangle _g)$ be a smooth $n$-dimensional Riemannian manifold without boundary, which represents the configuration space of a Lagrangian dynamical system. Elements in the tangent bundle $TM$ will be denoted by $(q, v)$, with $q \in M$ and $v \in T_qM$. Let $L: TM\to \mathbb{R}$ be a smooth autonomous (Lagrangian) function satisfying the following assumptions

${\bf(N1)}$ $L$ is non-degenerate on the fibers of $TM$, that is, for every $(q, v)\in TM$ we have that $d_{vv}L(q, v)\neq 0$ is non-degenerate as a quadratic form, where $d_{vv}L$ denotes the fiberwise second differential of $L$

${\bf(N2)}$ $L$ is exactly quadratic in the velocities meaning that the function $L(q, v)$ is a polynomial of degree at most $2$ with respect to $v$.

Remark 1.1. Before stating our main result, we observe that we require that the Lagrangian function is exactly quadratic in the velocity assumption (N2). The smoothness property of $L$ are in general not sufficient for guaranteeing the twice Fréchét differentiability of the Lagrangian action functional. In fact, the functional is twice Fréchét differentiable if and only if $L$ is exactly quadratic in the velocities. In this case, the Lagrangian action functional is actually smooth. This fact depends upon the differentiability properties of Nemitsky operators. We observe that, by using a finite dimensional approximations for the free-period action functional as developed by authors in ^[7], it should be possible to remove the technical condition (N2).

On the cartesian product $\Lambda^1(M)\times \mathbb{R}^+$, where $\Lambda^1(M)$ denotes the Hilbert manifold of $1$-periodic loops on $M$ having Sobolev regularity $H^1$, we define the free period Lagrangian action functional as

where $h$ is a real constant playing the role of energy. In fact, since the system is autonomous, the energy is a first integral. By a direct computation of the first variation of the action functional $\mathbb E_h$, it follows that $(x, T)$ is a critical point of $\mathbb E_h$ if and only if it satisfies the Euler-Lagrangian equations having fixed energy $h$.

Definition 1.2. Let $(x, T)$ be a critical point of $\mathbb E_h$. We term $(x, T)$ non-null if

Moreover, $(x, T)$ is termed

● $L$-Positive if $\langle d_{vv} L(x(t), x'(t))x'(t), x'(t) \rangle _g > 0$;

● $L$-Negative if $\langle d_{vv} L(x(t), x'(t))x'(t), x'(t) \rangle _g < 0$.

We observe that the notion of $L$-positivity (resp. $L$-negativity) provides a sort of generalization of the Legendre convexity (resp. concavity) condition only along the selected orbit. Recall that the classical Legendre convexity condition reads

${\bf(L1)}$ $L$ is $\mathscr C^2$ strictly convex on the fibers of $TM$, that is, for every $(q, v)\in TM$ we get $d_{vv}L(q, v) > 0$ as a quadratic form.

It is worth to observe that the above conditions have been introduced for avoiding any sign changing of the quanity

Otherwise, this provides deep difficulties, like in the definition of the spectral index.

Following authors, in [9,Definition 1.2], we are ready to introduce the notion of orbit cylinder.

Definition 1.3. A critical point $(x, T)$ of $\mathbb E_{h}$ admits an orbit cylinder if there exist $\epsilon > 0$ and a smooth (in $s$) family critical points $\{(x_{h+s}, T_{h+s}), s\in(-\epsilon, \epsilon)\}$ of $\mathbb{E}_{h+s}$ such that $(x_h, T_h) = (x, T)$. Moreover, this orbit cylinder is called non-degenerate if $T'(h)\neq0$.

Under the above notation our main result reads as follows.

Theorem 1.4. Let $(x, T)$ be a non-null critical point of the free-period Lagrangian action $\mathbb E_h$ admitting an orbit cylinder. If one of the following four statements holds:

(1) $x$ is $\bf{L-Positive}$ and

(OR) $x$ is orientation preserving and $\iota_{ \rm{spec}}(x, T) +n$ is even

(NOR) $x$ is orientation reversing and $\iota_{ \rm{spec}}(x, T) +n$ is odd

(2) $x$ is $\bf{L-Negative}$ and

(OR) $x$ is orientation preserving and $\iota_{ \rm{spec}}(x, T) +n$ is odd

(NOR) $x$ is orientation reversing and $\iota_{ \rm{spec}}(x, T) +n$ is even

then $x$ is linearly unstable.

Under the classical Legendre convexity condition, it is possible to prove that the spectral index reduces to the classical Morse index. So, we get the following.

COROLLARY 1. Let us assume that $(x, T)$ is a critical point of free period Lagrangian action $\mathbb E_h$ admitting an orbit cylinder and we assume that (L1) holds. If one of the following two alternatives holds

(OR) $x$ is orientation preserving and $\iota_{ {\mathrm{Mor}}}(x, T) +n$ is even

(NOR) $x$ is orientation reversing and $\iota_{ {\mathrm{Mor}}}(x, T) +n$ is odd

then $x$ is linearly unstable.

From a tecnical viewpoint a key step for proving our main result is based on a spectral flow formula relating the Morse index of a periodic orbit as critical point of the free and fixed time Lagrangian action functional. In ^[9], authors actually established such a relation under a non-degenerate assumption of the orbit cylider. As already observed, here we give a generalization of the aforementioned results provided in ^[9] by dropping the non-degeneracy assumption of the orbit cylinder. Such a generalization is essentially based on a quite new results constructed by the second named author in ^[10].

A crucial intermediate step for proving the main result of this paper is based on a spectral flow formula relating the spectral index to a symplectic invariant known in literature as Maslov index (which is an intersection invariant constructed in the Lagrangian Grassmannian manifold of a symplectic space) that plays a crucial role in detecting the stability properties of a periodic orbit. (For an index formula, we refer the interested reader to ^{[11,12,13,14]} and references therein). Very recently, new spectral flow formulas have been established and applied for detecting bifurcation of heteroclinic and homoclinic orbits of Hamiltonian systems or bifurcation of semi-Riemannian geodesics. (Cfr. ^[15,16,17]).

Notation

For the sake of the reader, let us introduce some common notation that we'll use henceforth throughout the paper.

● $(M, \langle{{\cdot}}, {{\cdot}} \rangle _g)$ denotes a Riemannian manifold without boundary, $TM$ its tangent bundle and $T^*M$ its cotangent bundle.

● $\Lambda^1(M)$ is the Hilbert manifold of loops on manifold $M$ having Sobolev regularity $H^1$.

● $\omega$ denotes the symplectic structure $J$ the standard symplectic matrix such that $\omega(\cdot, \cdot) = \langle J \cdot, \cdot \rangle$ where $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean product.

● $\iota_{ {\mathrm{Mor}}}(x)$ stands for the Morse index of $x$, $\iota_{ \rm{spec}}(x)$ for the spectral index of $x$, $\iota_{ \rm{geo}}(x)$ for the geometrical index of $x$, $\iota_{{1}}$ denotes the Maslov-type index or Conley-Zehnder index of a symplectic matrix path, $\iota_{ {\mathrm{CLM}}}$ denotes the Maslov (intersection) index and finally sf denotes the spectral flow.

● $P(L)$ denotes the set of $T$-periodic solutions of the Euler-Lagrange Equation, $P(H)$ the set of $T$-periodic solutions of the Hamiltonian Equation.

● $\delta_{ij}$ is the Kronecker symbol. $I_X$ or just $I$ will denote the identity operator on a space $X$ and we set for simplicity $I_k : = I_{ \mathbb{R}^k}$ for $k \in \mathbb{N}$. $\mathrm{Gr}(\cdot)$ denotes the graph of its argument, $\Delta$ denotes the graph of identity matrix $I$.

● $\mathbb{U}$ is the unit circle of the complex plane.

● $\mathrm{O}(n)$ denotes the orthogonal group, $\mathrm{Sp}(2n, \mathbb{R})$ or just $\mathrm{Sp}(2n)$ denotes the $2n\times 2n$ real symplectic group.

● $\mathfrak P$ denotes the linearized Poincaré map.

● $\mathcal{BF}^{sa}$ denotes the set of all bounded selfadjoint Fredholm operators, $\sigma(\cdot)$ denotes the spectrum of the operator in its argument.

● We denote throughout by the symbol ${{\cdot}}^{\mathrm{T}}$ (resp. ${{\cdot}}^{\mathrm{-T}}$) the transpose (resp. inverse transpose) of the operator $\cdot$. Moreover $\mathrm{rge}\, (\cdot), \ker(\cdot)$ and $\mathrm{rank}(\cdot)$ denote respectively the image, the kernel and the rank of the argument.

● $\Gamma$ denotes the crossing form and $\mathrm{n_+}$/$\mathrm{n_-}$ denote respectively the dimensions of the positive/negative spectral spaces and finally ${\rm{sgn}}(\cdot)$ is the signature of the quadratic form (or matrix) in its argument and it is given by ${\rm{sgn}}(\cdot) = \mathrm{n_+}(\cdot)- \mathrm{n_-}(\cdot)$.

2.   Lagrangian dynamics and variational framework

In this preliminary section we fix our basic notation and we explicitly provided the computation of the first and second variation of the free period Lagrangian action functional.

2.1.   Free-period Lagrangian action

Let $(M, \langle{{\cdot}}, {{\cdot}} \rangle _g)$ be a (not necessarily compact or connected) smooth $n$-dimensional Riemannian manifold without boundary, which represents the configuration space of a Lagrangian dynamical system and we denote by $\left\| {\cdot} \right\|$ the Riemannian norm. Elements in the tangent bundle $TM$ will be denoted by $(q, v)$, with $q \in M$ and $v \in T_qM$. The metric $\langle{{\cdot}}, {{\cdot}} \rangle _g$ induces: a metric on $TM$, Levi-Civita connections both on $M$ and $TM$ and finally the isomorphisms

for $T^v_{(q, v)}TM = \ker D\tau(q, v)$ where $\tau: TM \to M$ denotes the canonical tangent projection.

Notation 2.1. We shall denote by $\nabla_t$ the covariant derivative of a vector field along a smooth curve $x$ with respect to the metric $\langle{{\cdot}}, {{\cdot}} \rangle _g$. $\partial_q$ (resp. $\partial_v$) denotes the partial derivative along the horizontal part (resp. vertical part) given by the Levi-Civita connection in the splitting of $TTM$ and We shall denote by $\partial_{vv}, \partial_{qv}, \partial_{qq}$ the components of the Hessian in the splitting of $TTM$.

Given a positive number $T\in \mathbb{R}$, we denote by $\mathbb{T}$ the one-dimensional torus $\mathbb{T} = \mathbb{R}/T \mathbb{Z}$. Let $\Lambda^1_T(M)$ be the Hilbert manifold of all loops $y: \mathbb{T} \to M$ having Sobolev class $H^1$. Setting $x(t) = y(tT), t\in[0, 1]$ we get that the closed curve $y$ will be identified with the pair $(x, T)$. The action of $y$ on time interval $[0, T]$ is given by

where $h\in \mathbb{R}$ is a parameter. For convenience, we denote

In short-hand notation, in what follows we use $\Lambda^1(M)$ instead of $\Lambda^1_1(M)$. In this way, we define a one-to-one correspondente between

which preserves the action values. Bearing in mind such a correspondence, the free period action functional (2.1) is defined on the manifold $\Lambda^1(M)\times \mathbb{R}^+$.

It is well-known that the tangent space $T_x\Lambda^1(M)$ can be identified in a natural way with the Hilbert space of $1$-periodic $H^1$ (tangent) vector fields along $x$, i.e.,

It is worth noticing that under the (N2) assumption, the Lagrangian functional is of regularity class $\mathscr C^2$ (actually it is smooth). Let $\langle\langle \cdot, \cdot \rangle \rangle_1$ denote the Riemannian metric on $\Lambda^1(M)$ defined by

For $(\xi, b)\in \mathcal H(x)\times \mathbb{R}$, the first variation of $\mathbb E_h$ at $(x, T)\in\Lambda^1(M)\times \mathbb{R}^+$ is given by

and for $(0, b)\in \mathcal H(x)\times \mathbb{R}$, it reduces to

By Eqs (2.2) and (2.3), up to standard regularity arguments, it follows that critical points $(x, T)$ of $\mathbb E_h$ are $1$-periodic solutions of corresponding Euler-Lagrange equation having energy $h$. So, $(x, T)$ satisfies the following equations:

Now, being $\mathbb E_h$ smooth it follows that the first variation $d\mathbb E_h(x, T)$ at $(x, T)$ coincides with the Fréchét differential $D\mathbb E_h(x, T)$ and the second variation of $\mathbb E_h$ at $(x, T)$ coincides with $D^2 \mathbb E_h(x, T)$. We set

so, we get

Remark 2.2. For a complete details on the first and second variations, we refer the interested reader to ^[18] and references therein.

For a given critical point $(x, T)$ of $E_h$, let us consider the following fixed period Lagrangian action functional:

where $x\in \Lambda^1(M)$. Actually, $\mathbb E_{h}^T$ is the restriction of $\mathbb E_{h}$ to the submanifold $\Lambda^1(M)\times\{T\}$. By similar calculations, the first variation of $\mathbb E_{h}^T$ at $x\in\Lambda^1(M)$ is given by

Critical points $x$ of $\mathbb E_{h}^T$ are $1$-periodic solutions of corresponding Euler-Lagrange Equation:

The second variation of $\mathbb E_{h}^T$ at $x$ is given by

By the previous computation, we get that $\xi \in \ker d^2 \mathbb E_{h}^T(x)$ if and only if $\xi$ is a $H^2$ vector field along $x$ which solves weakly (in the Sobolev sense) the following boundary value problem

By standard bootstrap arguments, it follows that $\xi$ is also a classical (smooth) solution of Eq (2.6).

Remark 2.3. It is easy to prove that $x$ is a critical point of fixed period Lagrangian system (2.4) provided that $(x, T)$ is a critical point of free period Lagrangian system.

The next result provides an answer to the existence of an orbit cylinder about a $T$-periodic orbit $x$. We refer the interested reader to [19,Section 4.1,Proposition 2] for the proof.

Proposition 2.4. Let us assume that a periodic solution $x(t, E^*)$ of a Hamiltonian vector field $X_H$ on $M$ having energy $E^* = H(x(t, E^*))$ and period $T^*$ has exactly two Floquet multipliers equal $1$. Then, there exists a unique and smooth one-parameter family $x(t, E)$ of periodic solutions with periods $T(E)$ close to $T^*$ and lying on the energy surfaces $H(x(t, E)) = E$ for $|E-E^*|$ sufficiently small. Moreover, $T(E)$ converges to $T(E^*)$ for $E\rightarrow E^*$.

3.   Spectral index, geometrical index and Poincaré map

The goal of this section is to associate at the critical point $(x, T)$ of the free period Lagrangian action given at Eq (2.1) and to a critical point $x$ of the fixed period Lagrangian action given at Eq (2.4) the spectral index, defined in terms of the spectral flow of a suitable path of Fredholm quadratic forms. We refer the interested reader to [20,Appendix B] and references therein for a discussion about the spectral flow.

3.1.   Spectral index: an intrinsic (coordinate free) definition

Given $(x, T)$ be a critical point of $\mathbb E_{h}$, for any $s \in [0, +\infty)$ we let $\mathcal I_s: (\mathcal H(x)\times \mathbb{R})\times (\mathcal H(x)\times \mathbb{R}) \to \mathbb{R}$ the bilinear form defined by

Notation 3.1. In short-hand notation and if no confusion can arise, we set $\mathcal Q^h\ = d^2 \mathbb E_{h}(x, T)$.

Proposition 3.2. For any $s \in [0, +\infty)$ let $\mathcal Q_s$ denote the quadratic form associated to $\mathcal I_s$ defined at Eq (3.1). Then

$s\mapsto \mathcal Q_s$ is a smooth path of Fredholm quadratic forms onto $\mathcal H(x)\times \mathbb{R}$. In particular $\mathcal Q^h$ is a Fredholm quadratic form on $\mathcal H(x)\times \mathbb{R}$.

Proof. The proof is completely analogous to [20,Proposition 3.2]. We let $(x, T)$ be a critical point of the Lagrangian action functional (2.1). Then we can define the bilinear form $\mathcal I_{s, T}: \mathcal H(x)\times \mathcal H(x) \to \mathbb{R}$ of $x$ as a critical point of system (2.4) in the same way as $(x, T)$. In fact, $\mathcal I_{s, T}$ is given by

Set $\mathcal Q^h_T\ = d^2 \mathbb E_{h}^T(x)$. Arguing precisely as done in Proposition 3.2, it can be proved $\mathcal I_{s, T}$ and $\mathcal Q^h_T$ both are Fredholm quadratic forms.

For any $s \in [0, +\infty)$, let $\mathcal Q_{s, T}$ denote the quadratic form associated to $\mathcal I_{s, T}$. Now we are entitled to define the following spectral indexes.

Definition 3.3. Let $(x, T)$ be a non-null critical point of the free period Lagrangian action given at Eq (2.3). We term spectral indices of $(x, T)$ and $x$ are respectively the integers given by

where the (RHS) denotes the spectral flow of the path of Fredholm quadratic forms defined on the interval $[0, s_0]$ for a sufficiently arge $s_0 > 0$.

Remark 3.4. It is easy to show that Definition 3.3 is well-posed meaning that the spectral indices are independent on $s_0$. This fact will be proved in the sequel and it is actually a direct consequence of Lemma 3.6.

Proposition 3.5. We assume that (L1) holds. Then the Morse indices of $(x, T)$ and $x$ (i.e., the dimension of the maximal negative subspace of the Hessian of $\mathcal Q^h$ and $\mathcal Q^{h_T}$) are both finite and the following equalities hold

Proof. We only consider the free period Lagrangian action, being the fixed period case, completely analogous. So, we start by observing that if $L$ is $\mathscr C^2$-strictly convex on $TM$, then $\alpha(x, T)$ is a positive Fredholm quadratic form and hence $s\mapsto \mathcal Q_s$ is a path of essentially positive Fredholm quadratic forms being realized by a path of compact symmetric perturbation of a positive definite Fredholm operator. In particular the Morse index of $\mathcal Q_s$ is finite for every $s \in [0, +\infty)$. If $s_0$ is sufficiently large the form $\mathcal Q_{s_0}$ is non-degenerate and positive definite being the quadratic form associated to $\alpha(x, T)$ be Fredholm and positive definite. In particular, its Morse index vanishes. Since the spectral flow for path of essentially positive Fredholm quadratic forms is equal to the difference of the Morse indices at the ends (cf. [20,Appendix B]), we get that

This concludes the proof.

3.2.   Pull-back bundles and push-forward of Fredholm forms

We denote by ${\bf{E}}$ the $\langle{{\cdot}}, {{\cdot}} \rangle _g$-orthonormal and parallel frame pointwise given by

Given a critical point $(x, T)$ of the free period Lagrangian action, we let $\bar A: T_{x(0)}M\to T_{x(1)}M\cong T_{x(0)}M$ the $\langle{{\cdot}}, {{\cdot}} \rangle _g$-orthogonal operator defined by

Such a frame ${\bf{E}}$, induces a trivialization of the pull-back bundle $x^*(TM)$ over $[0, 1]$ through the smooth curve $x:[0, 1] \to M$; namely the smooth one parameter family of isomorphisms

here $\{e_i\}_{i = 1}^n$ is the canonical basis of $\mathbb{R}^n$ and $\delta_{ij}$ denotes the Kronecker symbol.

By Eq (3.3) we get that the pull-back by $E_t$ of the metric $\langle{{\cdot}}, {{\cdot}} \rangle _g$ induces the Euclidean product on $\mathbb{R}^n$ and moreover this pull-back is independent on $t$, as directly follows by the orthogonality assumption on the frame ${\bf{E}}$.

We set $A\ = E_0^{-1}\bar A^{-1} E_1 \in \mathrm{O}(n)$ and define

Let us now consider the Hilbert space

equipped with the inner product

Denoting by $\Psi: \mathcal H(x) \to H^1_A([0, 1], \mathbb{R}^n)$ the map defined by $\Psi(\xi) = u$ where $u(t) = E_t^{-1}(\xi(t))$, it follows that $\Psi$ is a linear isomorphism and it is easy to check that

where, in the last, we used property of the frame being parallel.

For $i = 1, \dots, n$ and $t \in[0, 1]$, we let $e_i(t)\ = E_t e_i$ and we denote by $\langle P(t) \cdot, \cdot \rangle$, $\langle Q(t) \cdot, \cdot \rangle$ and $\langle R(t) \cdot, \cdot \rangle$ respectively the pull-back by $E_t$ of $\langle{{\bar P(t) \cdot}}, {{ \cdot }} \rangle _g$, $\langle{{ \bar Q(t) \cdot}}, {{\cdot }} \rangle _g$ and $\langle{{ \bar R(t) \cdot}}, {{\cdot }} \rangle _g$ and y $L$ the pull-back of $\bar L$ by $E_t$. Thus, we get

We observe that $P$ and $R$ are symmetric matrices and being $e_i(T) = \sum_{j = 1}^na_{ij}e_j(0)$ we infer that

Now, for every $s \in [0, +\infty)$, the push-forward by $\Psi$ of the index forms $\mathcal I_s$ on $\mathcal H(x)\times \mathbb{R}$ is given by the symmetric bilinear forms on $H^1_A([0, 1], \mathbb{R}^n)\times \mathbb{R}$ defined by

Denoting by $q^A_s$ the quadratic form on $H^1_A([0, 1], \mathbb{R}^n)\times \mathbb{R}$ associated to $I_s$ then, as direct consequence of Proposition 3.2, we get that for every $s \in [0, +\infty)$, the quadratic form $q_s^A$ is Fredholm on $H^1_A([0, 1], \mathbb{R}^n)\times \mathbb{R}$. The following result is crucial in the well-posedness of the spectral index.

Lemma 3.6. Under the above notation, there exists $s_0 \in [0, +\infty)$ large enough such that for every $s \ge s_0$, the form $I_s$ given in Eq (3.6) is non-degenerate (in the sense of bilinear forms).

Proof. We argue by contradiction and we assume that for every $s_0 \ge 0$ there exists $s \ge s_0$ such that $I_s$ is degenerate. Then there exists a $(u, b)\in H^1_A([0, 1], \mathbb{R}^n)\times \mathbb{R}$ such that $I_s((u, b), (v, d))\equiv 0$ for every $(v, d)\in H^1_A([0, 1], \mathbb{R}^n)\times \mathbb{R}$, namely we have

We let $v(t)\ = \frac 1 TP(t)u(t)$ and we observe that as direct consequence of Eq (3.5), the function $v$ is admissible (meaning that $v$ belongs to $H^1_A$). Since $(x, T)$ is non-null, we set $d\ = {\rm{sign }} \kappa(t)\cdot b$ for ${\rm{sign }} \kappa(t)\in\{1, -1\}$.

By replacing $v$ and $d$ into Eq (3.7) respectively by $v = Pu/T$ and $d = {\rm{sign }} \kappa(t)\cdot b$ and dropping the argument $t$ in each function, we get

For $i = 1, 2, 3, 4$, we let $C_i$ be given as follows:

and for for $s$ sufficiently large we get

This concludes the proof.

Remark 3.7. Here we would like to observe that the ASSUMPTION has been strongly used in the proof of Lemma 3.6 and it is crucial in the previous construction. However, it is interesting to understand if this condition is just technical or if it represent an obstruction to carry over this case Theorem 1.4.

Now, for every $s \in [0, +\infty)$, the push-forward by $\Psi$ of the index form $\mathcal I_s$ on $\mathcal H(x)$ is given by the symmetric bilinear form on $H^1_A([0, 1], \mathbb{R}^n)$ defined by

By arguing precisely as before, we get that there exists a $s_0 > 0$ large enough such that $I_{s, T}$ is non-degenerate for every $s > s_0$.

Proposition 3.8. Let $(x, T)$ be a non-null critical point of Lagrangian action given at Eq (2.1). Then the spectral indexes are well-defined.

Proof. We start observing that, $s\mapsto q^A_s$ is a path of Fredholm quadratic forms on $H^1_A([0, T], \mathbb{R}^n)\times \mathbb{R}$. Moreover, by Lemma 3.6, there exists $s_0 \in [0, +\infty)$ such that $q^A_s$ is non-degenerate for every $s \ge s_0$ and hence the integer ${\rm{sf }}\{q^A_s, s \in [0, s_0]\}$ is well-defined.

The conclusion follows by observing that $q^A_s$ is the push-forward by $\Psi$ of the Fredholm quadratic form $\mathcal Q_s$ and by the fact that the spectral flow of a generalized family of Fredholm quadratic forms on the (trivial) Hilbert bundle $[0, s_0] \times (\mathcal H(x)\times \mathbb{R})$ is independent on the trivialization. This concludes the proof.

3.3.   The difference between two spectral indices

This subsection is to provide an abstract formula for computing the difference between the spectral indices defined above.

Let $H$ be a real separable Hilbert space equipped with inner product $\langle\cdot, \cdot\rangle$, $W\subset H$ be a dense subspace and let the inclusion map $i:W\rightarrow H$ be compact. We let $A$ be an unbounded linear operator on $H$ having domain $W$ and we assume that $A$ is a Fredholm operator. Given a finite dimensional Hilbert space $V$, we assume that $B:V\rightarrow H$ is a bounded linear operator and $C:V\rightarrow V$ is a bounded self-adjoint linear operator. We denote by $\mathcal A:W\oplus V\rightarrow H\oplus V$ the self-adjoint operator defined by

where $B^*$ is the adjoint operator of $B$. In matrix form the operator $\mathcal A$ can be written as

Lemma 3.9. Under above assumptions, we have

where $m^-$ denotes the Morse index.

Proof. For the Hilbert space $V$, the following splitting holds

By choosing a suitable basis, the matrix $\left[ {\begin{array}{*{20}{c}} 0&B\\B^*&C \end{array}} \right]$ has the block form

where $B_{11}: \mathrm{Im}\, B^*\rightarrow \mathrm{Im}\, B$ and $B^*_{11}: \mathrm{Im}\, B\rightarrow \mathrm{Im}\, B^*$ are both invertible. So, in particular

Now, since

the thesis readily follows.

Lemma 3.10. For $s\in[0, 1]$, we let $\mathcal A(s) = \left[ {\begin{array}{*{20}{c}} A & (1-s)B\\(1-s)B^* & (1-s)C \end{array}} \right]$ Then, the following spectral flow formula holds

Proof. We start to consider the splitting $W = (\ker A)^\bot\oplus\ker A$. So, $\mathcal A(s)$ can be written in the following block form

where $A_{11}:(\ker A)^\bot\rightarrow (\ker A)^\bot$ is invertible and $B_1: V\rightarrow (\ker A)^\bot, B_2:V\rightarrow \ker A$.

For $s\in[0, 1]$, we let

First claim. The following equality holds:

This equality is a direct consequence of the stratum homotopy invariant property of the spectral flow. So, let's start to consider the $2$-parameter family of operators pointwise defined by

and we observe that we have $A(s, t) = K(t)^* A(s) K(t)$ for

By a straightforward calculation it follows that $\dim \ker A(0, t)$ and $\dim \ker A(1, t)$ are both constants. By using the zero axiom of the spectral flow (namely each path is contained in a fixed stratum of the Fredholm Lagrangian Grassmannian), we get that

By invoking the stratum homotopy invariant property of the spectral flow, we get that

which is precisely the equality appearing at Eq (3.9).

Let

By taking into account the additivity property of the spectral flow under direct sum as well as of Eq (3.9), we get that

Now, since $\mathcal C(1) = 0$, then we have

By Lemma 3.9, we get that

In order to conclude, we have to prove that

Let us consider ${{(x_1, x_2, 0)}}^\mathit{T}\in \ker \mathcal A(0)\cap W$. Then for every ${{(u_1, u_2, v)}}^\mathit{T}\in W\oplus V$ we have

We set $v = 0$. So, $\langle A_{11}x_1, u_1 \rangle\equiv0$ for every $u_1\in(\ker A)^{\bot}$ implies that $A_{11}x_1 = 0$. Consequently we have $x_1 = 0$. Now, Eq (3.10) becomes $\langle B_2^*x_2, v \rangle\equiv 0$. Since $v$ is arbitrary, then $B_2^*x_2 = 0$. Hence $W\cap \ker \mathcal A(0) = \{ {{(0, x_2, 0)}}^\mathit{T}\ | \ B_2^*x_2 = 0\} = \ker B_2^*$.

If ${{(x_1, x_2, y)}}^\mathit{T}\in W^{\bot}$, then for every ${{(u_1, u_2, 0)}}^\mathit{T}\in W$ we have

We set $u_2 = 0$. Then $\langle A_{11}x_1+B_1y, u_1 \rangle\equiv0$ for every $u_1\in(\ker A)^{\bot}$ implies that $A_{11}x_1+B_1y = 0$. Consequently we have $x_1 = -A_{11}^{-1}B_1y$. Let $u_1 = 0$, Eq (3.11) becomes $\langle B_2y, u_2 \rangle\equiv 0$ for every $u_2\in\ker A$, then $B_2y = 0$. Hence $W^{\bot} = \{ {{(-A_{11}^{-1}B_1y, x_2, y)}}^\mathit{T}\ | \ B_2y = 0, x_2\in\ker A\}$ and $W\cap W^{\bot} = \{ {{(0, x_2, 0)}}^\mathit{T}\} = \ker A$.

Now, we get

For every $\xi_0 = {{(-A_{11}^{-1}B_1y, x_2, y)}}^\mathit{T}\in W^{\bot}$, we have

Therefore, we have $m^{-}(\mathcal A(0)|_{W^{\bot}}) = m^-((C-B_1^*A_{11}^{-1}B_1)|_{\ker B_2})$. This concludes the proof.

Lemma 3.11. We let $\mathcal A(s) = \left[ {\begin{array}{*{20}{c}} A&sB\\sB^*&sC \end{array}} \right]$, for $s\in[0, 1]$ and we assume that $A$ is invertible. Then we have

Proof. By a similar discussion as provided in the proof of Eq (3.9), we get

Since $A$ is invertible, we infer that

We observe that the operator $s(C-sB^*A^{-1}B)$ is defined on a finite dimensional space $V$, and since in this case the spectral flow is equal to the Morse index at the starting point minus the Morse index at the end point, we get that

By using the previous results, we are now ready to compute the difference between two spectral indices defined in Definition 3.3. By taking into account Eq (3.6), we denote by $\mathcal A(s)$ the realization of $I_s$ meaning the bounded linear operator representing the bilinear form $I_s$ w.r.t. the $H^1_A\times \mathbb{R}$ scalar product; so, we get

Similarly, we define the bounded linear operators $A(s), B, C(s)$ representing w.r.t. the $H_A^1 \times \mathbb{R}$ scalar product the three terms defining $I_s$. So, we get

In matrix form the operator $\mathcal A(s)$ can be written as

Let now consider the homotopy pointwise defined by

Recall the discussions below Remark 3.7 we proved that the index form $I_{s, T}$ is non-degenerate for $s_0$ large enough. Therefore, $A(s_0)$ is invertible. The next result provides a more strinking propery about the spectrum of $A(s_0)$.

Lemma 3.12. There exists $\delta > 0$ such that $\sigma(A(s_0))\cap[-\delta, \delta] = \emptyset$. In particular, the operator $A^{-1}(s_0)$ is bounded.

Proof. Arguing by contradiction, we assume that for every $\delta > 0$ there exists $\lambda_\delta\in[-\delta, \delta]$ and $u_\delta\in H^1_A([0, 1], \mathbb{R}^n)$ such that $A(s_0)u_\delta = \lambda_\delta u_\delta$. Take $v_\delta = Pu_\delta$, then we have

By taking into account Eq (3.8), we have

Inequalities provided at Eq (3.13) and Eq (3.14) contradict each other for $s_0$ sufficiently large and $\delta$ (consequently $\lambda_\delta$) sufficiently small. This concludes the proof.

By the homotopy invariance property of the spectral flow, we get that

Let us now compute the spectral flow ${\rm{sf }}(\mathcal A(s_0, 1-\epsilon), \epsilon\in[0, 1])$. By using Lemma 3.12, we infer that $B^*A^{-1}(s_0)B$ is a bounded operator (on a one-dimensional space). So, we get

By Lemma 3.11, we have

Let us now compute the spectral flow ${\rm{sf }} (\mathcal A(0, \epsilon), \epsilon \in [0, 1])$. By using Lemma 3.10 we have

where $W = H^1_A([0, 1], \mathbb{R}^n)$ and $V = \mathbb{R}$.

The next step is to provide an explicit description of

The basic idea comes from [9,Section 2.1].

Let $(x, T)$ be a non-null critical point of $\mathbb E_{h}$ with orbit cylinder $(x_{h+s}, T_{h+s})$. Then for every $(\xi, b)\in\mathcal H(x)\times \mathbb{R}$ we have

By differentiating w.r.t. $s$ both sides of Eq (3.17), we get

where $\xi_h(t) = \frac{\partial}{\partial s}|_{s = 0}x_{h+s}(t), T'(h) = \frac{d}{ds}|_{s = 0}T_{h+s}.$ Let now choose a variation $\{(x_{h, r}, T_{h, r}), r\in(-\epsilon, \epsilon)\}$ such that $(x_{h, 0}, T_{h, 0}) = (x, T)$ and $\frac{\partial}{\partial r}|_{r = 0}(x_{h, r}, T_{h, r}) = (\xi, b)$, then we have

By taking into account Eq (3.18) we have

for every $(\xi, b)\in \mathcal H(x)\times \mathbb{R}$. Taking $b = 0$ and $(\xi, b) = (\xi_h, T'(h))$ respectively, we have

Let us identify $\mathcal H(x)$ with $\mathcal H(x)\times \{0\}$ and we denote the Hessian of $\mathbb E_{h}$ by $\nabla^2 \mathbb E_{h}(x, T)$. So, $\ker d^2 \mathbb E_{h}(x, T) = \ker \nabla^2 \mathbb E_{h}(x, T)$. We now set

The following result holds.

Lemma 3.13. Under above notations, we get

Proof. We argue by contradiction. If $\mathcal H(x)+\ker d^2 \mathbb E_{h}(x, T) = \mathcal H(x)\times \mathbb{R}$, then

By taking into account Eq (3.20) we get $(\xi_h, T'(h))\in\mathcal H^\bot(x)$ and by using Eq (3.19) we know $(\xi_h, T'(h))\notin\ker d^2 \mathbb E_{h}(x, T)$ which contradicts Eq (3.21).

So, $\mathcal H(x)+\ker d^2 \mathbb E_{h}(x, T)\neq\mathcal H(x)\times \mathbb{R}$. Since $\dim((\mathcal H(x)\times \mathbb{R})/\mathcal H(x)) = 1,$ then we finally get $\ker d^2 \mathbb E_{h}(x, T)\subset \mathcal H(x)$.

Now observe that $\ker d^2 \mathbb E_{h}(x, T)\oplus \mathbb{R}(\xi_h, T'(h))\subset \mathcal H^\bot(x)$ and $\dim(\mathcal H^\bot(x)/\ker d^2 \mathbb E_{h}(x, T))\leq1$. In particular $\mathcal H^\bot(x) = \ker d^2 \mathbb E_{h}(x, T)\oplus \mathbb{R}(\xi_h, T'(h))$. This concludes the proof.

By invoking Lemma 3.13, we get

If $T'(h)\neq0$, the $\mathbb{R}(\xi_h, T'(h))\subsetneq \mathcal H(x)$. Thus, we have $\mathcal H(x)\cap\mathcal H^\bot(x) = \ker d^2 \mathbb E_{h}(x, T)$.

At the same time, by Lemma 3.13 we have $d^2 \mathbb E_{h}(x, T)|_{\mathcal H^\bot(x)} = d^2 \mathbb E_{h}(x, T)|_{ \mathbb{R}(\xi_h, T'(h))}$. Then by Eq (3.20), we have

Hence, we have

If $T'(h) = 0$, then by Eq (3.20), we get that $(\xi_h, T'(h))\in \mathcal H(x)\cap\mathcal H^\bot(x)$. Therefore, we have

As a result, we have $\dim(\mathcal H(x)\cap\mathcal H^\bot(x))-\dim(\mathcal H(x)\cap \ker d^2 \mathbb E_{h}(x, T)) = 1$, and

So, we get that if $T'(h) = 0$ there holds

Summing up, we have

In conclusion, we get

where $W = H^1_A([0, 1], \mathbb{R}^n)$. By summarizing all the previous results, the following theorem holds.

Theorem 3.14. Under above notations the following equalities hold:

Proof. The proof readily follows by invoking Eqs (3.15)-(3.16) and Eq (3.22). This concludes the proof.

Remark 3.15. We observe that the main role of orbit cylinder is to ensure the existence of vector $(\xi_h, T'(h))$ in Eq (3.19). A natural problem is to find out some more general conditions to insure the existence of a vector in $\mathcal H^\bot(x)$ but not in $\ker d^2 \mathbb E_{h}(x, T)$. The bifurcation theory of Hamiltonian system could be the right direction for answering this question.

By using Proposition 3.5, the following result holds.

Corollary 3.16. If $L$ is $\mathscr C^2$-strictly convex on $TM$, then the difference between two Morse indices is given by

Proof. Since $L$ is $\mathscr C^2$-strictly convex, then $\kappa = \langle Px', x'\rangle > 0$. By Proposition 3.5 and Eq (3.23), we conclude the proof.

Remark 3.17 Here we point out that if $(x, T)$ is a minimizer of system (2.4) when $L$ is $\mathscr C^2$-strictly convex, then we must have $\iota_{ {\mathrm{Mor}}}(x, T) = 0$. By Corollary 3.16 there must hold $T'(h) < 0$. Or vice versa, if $T'(h)\ge 0$, then $(x, T)$ cannot be a minimizer.

4.   Linear instability and proof of the main result

In this section we recall some well-known results about the fixed period problem. We refer the interested reader to ^[20] for the complete proofs.

4.1.   Hamiltonian system and geometrical index

It is well-known that under the assumption (N1) the Legendre transform

is a local smooth diffeomorphism. The Fenchel transform of $L$ is the autonomous Hamiltonian on $T^*M$

for every $(q, p) \in T^*M$, where the map $v$ is a component of the fiber-preserving diffeomorphism

the inverse of $\mathscr L_L$.

By the above Legendre transform, the Euler-Lagrange Equation (2.5) is changed into the following Hamiltonian equation:

By trivializing the pull-back bundle $x^*(TM)$ over $TM$ through the frame ${\bf{E}}$ defined in Subsection 3, Eq (2.6) is changed into

By setting $y(t) = \frac 1 TP(t) u'(t)+Q(t)u(t)$ and $z(t) = {{(y(t), u(t))}}^\mathit{T}$ we finally get

and $A_d$ has been defined in Eq (3.4).

In the standard symplectic space $(\mathbb{R}^{2n}, \omega)$, we denote by $J$ the standard symplectic matrix defined by $J = \left[ {\begin{array}{*{20}{c}} 0 & - I\\ I & 0\end{array}} \right]$. Thus the symplectic form $\omega$ can be represented with respect to the Euclidean product $\langle\cdot, \cdot\rangle$ by $J$ as follows $\omega(z_1, z_2) = \langle J z_1, z_2\rangle$ for every $z_1, z_2 \in \mathbb{R}^{2n}$.

Given $M \in \mathrm{Sp}(2n, \mathbb{R})$, we denote by $\mathrm{Gr}(M) = \{(x, Mx)|x\in \mathbb{R}^{2n}\}$ its graph and we recall that $\mathrm{Gr}(M)$ is a Lagrangian subspace of the symplectic space $(\mathbb{R}^{2n} \times \mathbb{R}^{2n}, -\omega \times \omega)$.

Definition 4.1. Let $x$ be a $1$-periodic solution of Eq (2.5), $z_x$ be the solution of corresponding Hamiltonian equation and let us consider the path

We define the geometrical index of $x$ as follows

where the (RHS) in Eq (4.3) denotes the $\iota_{ {\mathrm{CLM}}}$ Maslov index between the Lagrangian path $t \mapsto \mathrm{Gr}(\gamma_\Phi(t))$ and the Lagrangian path $\Delta\ = \mathrm{Gr}(I)$ defined at Appendix A and references therein.

Let $x$ be a $1$-periodic solution of Eq (2.5) and $z_x$ be the solution of corresponding Hamiltonian equation, we can define the linearized Poincaré map of $z_x$ as follows.

where $\zeta$ is the unique vector field along $x$ such that $\zeta(0) = \alpha_0$ and $\frac 1 T\bar P(0) \nabla_t \zeta(0)+ \bar Q(0)\zeta(0) = \delta_0$. Fixed points of $\mathfrak P_{z_x}$ correspond to periodic vector fields along $z_x$.

By pulling back the linearized Poincaré map defined in Eq (4.2) through the unitary trivialization $\Phi^ {\bf{E}}$ of $z_x(TT^*M)$ over $[0, 1]$ we get the map

where $z(t) = \big(y(t), u(t)\big)$ is the unique solution of the Hamiltonian system given in Eq (4.1) such that $z(0) = (y_0, u_0)$.

Denoting by $t \mapsto\psi(t)$ the fundamental solution of (linear) Hamiltonian system given in Eq (4.1), then we get the geometrical index given in Definition 4.1 reduces to

Moreover, the linearized Poincaré map can be given by the symplectic matrix $A_d\psi(1)$.

By choosing a suitable coordinates and trivialization we can split $A_d\psi(t)$ into following form:

where $P_x(t)$ is a path of $2(n-1)\times 2(n-1)$ symplectic matrices. It is referred to [9,Page 104-105].

Definition 4.2. Under above notations, $z_x$ is termed spectrally stable if the spectrum $\sigma({P_{x}(1)}) \subset \mathbb{U}$ where $\mathbb{U}\subset \mathbb{C}$ denotes the unit circle of the complex plane. If $P_{x}(1)$ is also semi-simple, then $z_x$ is termed linearly stable.

Denote $\gamma_1(t) = \Big\{\left[ {\begin{array}{*{20}{c}} 1 & 0\\-tT'(h) & 1 \end{array}} \right], t\in[0, 1]\Big\}$ and $\gamma_2(t) = \{P_x(t) \ | \ t\in[0, 1] \}$, then by invoking Eq (6.3), then we get

The, next result is well-known and relates the parity of the $\iota_{ {\mathrm{CLM}}}$-index to the linear instability of the periodic orbit.

Lemma 4.3. The following implication holds

Proof. It is referred to the proof of [20,Lemma 3.15].

In [20,Equation 4.25] we give the precise relationship between $\iota_{ \rm{geo}}(x)$ and $\iota_{ \rm{spec}}^T(x)$:

Remark 4.4. We conclude this section, by observing that even if not explicitly stated, all arguments provided above, also work when the periodic orbit is transversally degenerate.

4.2.   Proof of Main Theorem

Proof of Theorem 1.4. We prove only the (contrapositive of) the first statement in Theorem 1.4, being the others completely analogous. Thus, we aim to prove that

First of all, we have

Being $x$ orientation preserving (by assumption), then $\det A = 1$ and being $A$ also orthogonal, then we get that

● $n \;{\rm{ even }}\; \Rightarrow \dim\ker(A- I) \rm{ even}$

● $n \;{\rm{ odd }}\; \Rightarrow \dim\ker(A- I) \rm{ odd}$.

So, in both cases we have $n-\dim\ker(A- I)$ is even. Since $x$ is $L$-Positive, then $\kappa(t) > 0$. By Equations (3.23) and (4.4), we have

● $T'(h) \ge 0 \Rightarrow \iota_{ \rm{spec}}(x, T)- \iota_{ \rm{spec}}^T(x) \rm{ odd }$ and $\iota_{ \rm{geo}}(x)- \iota_{ {\mathrm{CLM}}}(\Delta, \mathrm{Gr}(\gamma_2(t))) \rm{ even }$

● $T'(h) < 0 \Rightarrow \iota_{ \rm{spec}}(x, T)- \iota_{ \rm{spec}}^T(x) \rm{ even}$ and $\iota_{ \rm{geo}}(x)- \iota_{ {\mathrm{CLM}}}(\Delta, \mathrm{Gr}(\gamma_2(t))) \rm{ odd }$.

So, in both cases we have $(\iota_{ \rm{spec}}(x, T)- \iota_{ \rm{spec}}^T(x))+(\iota_{ \rm{geo}}(x)- \iota_{ {\mathrm{CLM}}}(\Delta, \mathrm{Gr}(\gamma_2(t))))$ is odd. If $x$ is linear stable, then by taking into account Lemma 4.3, we get that $\iota_{ {\mathrm{CLM}}}(\Delta, \mathrm{Gr}(\gamma_2(t)))$ is even. Then, by Eq (4.14), we finally get that $n+ \iota_{ \rm{spec}}(x, T)$ is odd. This concludes the proof.

5.   A classical example

In this section, we will give a simple example where $T'(h) = 0$ inspired by [9,Section 5].

Let $(r, \theta)$ denote the polar coordinate on $\mathbb{R}^2$. Suppose $D\ = \{ (r, \theta)\ |\ 0 < r < 4\}$ and $f(r) = -\frac 1 2(r^3-4r^2+3r)$, then define $L: TD\rightarrow \mathbb{R}$ by

The energy function $E:TD\rightarrow \mathbb{R}$ is given by $E(r, \theta, \dot{r}, \dot{\theta}) = \frac 1 2(\dot{r}^2+r^2\dot{\theta}^2)$.

By a direct computation we have $\gamma(t)\ = (r(t), \theta(t))$ satisfies the Euler-Lagrange equations if and only if

From now on we only consider the circular orbit. Suppose

where $\rho(k)$ and $a(k)$ are both positive constants. If $\gamma(t)$ is an orbit, since the second equation of (5.1) is automatically satisfied, then we only requir

The energy $E = \frac 1 2 (\rho^2(k)a^2(k)) = k,$ then we have

We note that $f'(r) = (3r^2-8r+3)/2$, then by Equations (5.2)-(5.3), we have $\sqrt{2k} = -(3\rho^2(k) -8\rho(k)+3)/2$, namely,

By solving the above equation, then we get $\rho(k) = \frac 4 3-\sqrt{\frac 7 9-\frac 2 3\sqrt{2k}}$.

Consider $k\in(\frac 1 2-\epsilon, \frac 1 2+\epsilon)$, where $\epsilon$ is a sufficiently small positive number. For $k = \frac 1 2,$ it is easy to calculate that

By using Eq (5.3), then we have $T(k) = \dfrac{2\pi}{a(k)} = \dfrac{2\pi\rho(k)}{\sqrt{2k}},$ then

Therefore, $T'(1/2) = 2\pi(\rho'(1/2)-\rho(1/2)) = 0.$ By Corollary 3.16, then we get

A.   A symplectic excursion on the Maslov index

The purpose of this section is to provide the symplectic preliminaries used in the paper. In Subsection A.1, we give the definition of the Maslov index. Then we compute the Maslov index of a special symplectic path. Our basic references are ^{[13,14,21,22]}.

A.1.   A quick recap on the $\iota_{ {\mathrm{CLM}}}$-index

Given a $2n$-dimensional (real) symplectic space $(V, \omega)$, a Lagrangian subspace of $V$ is an $n$-dimensional subspace $L \subset V$ such that $L = L^\omega$ where $L^\omega$ denotes the symplectic orthogonal, i.e., the orthogonal with respect to the symplectic structure. We denote by $\Lambda = \Lambda(V, \omega)$ the Lagrangian Grassmannian of $(V, \omega)$, namely the set of all Lagrangian subspaces of $(V, \omega)$

It is well-known that $\Lambda(V, \omega)$ is a manifold. For each $L_0 \in \Lambda$, let

Each $\Lambda^k(L_0)$ is a real compact, connected submanifold of codimension $k(k+1)/2$. The topological closure of $\Lambda^1(L_0)$ is the Maslov cycle that can be also described as follows

The top-stratum $\Lambda^1(L_0)$ is co-oriented meaning that it has a transverse orientation. To be more precise, for each $L \in \Lambda^1(L_0)$, the path of Lagrangian subspaces $(-\delta, \delta) \mapsto e^{tJ} L$ cross $\Lambda^1(L_0)$ transversally, and as $t$ increases the path points to the transverse direction. Thus the Maslov cycle is two-sidedly embedded in $\Lambda(V, \omega)$. Based on the topological properties of the Lagrangian Grassmannian manifold, it is possible to define a fixed endpoints homotopy invariant called Maslov index.

Definition A.1. Let $L_0 \in \Lambda(V, \omega)$ and let $\ell:[0, 1] \to \Lambda(V, \omega)$ be a continuous path. We define the Maslov index $\iota_{ {\mathrm{CLM}}}$ as follows:

where the right hand-side denotes the intersection number and $0 < \varepsilon < < 1$.

For further reference and properties of the Maslov index we refer the interested reader to ^[21] and references therein.

For the special symplectic path $\gamma(t) = \left[ {\begin{array}{*{20}{c}} M_{11}(t) & 0\\M_{21}(t)& M_{22}(t) \end{array}} \right], t\in[0, T]$, there is a very useful formula to compute its Maslov index [23,Theorem 2.2]. Here we only give the simplified version. Let $V$ be a subspace of $\mathbb{C}^{2n}$, define

Then there holds

where $m^+$ denotes the Morse positive index and $S(t) = \{x\in \mathbb{C}^n\ | \ {{(x, M_{1, 1}x)}}^\mathit{T}\in V^I\}$. Please note that in our situation we take $K, R$ in [23,Theorem 2.2] as $I$ and $V$ respectively.

Take $V = \{{{(x, x)}}^\mathit{T}\ | \ x\in \mathbb{R}\}$, then $V^I = V$ and $\Delta\ = Gr(I) = W_I(V)$. Let $\gamma(t) = \Big\{\left[ {\begin{array}{*{20}{c}} 1 & 0\\tT_0 & 1 \end{array}} \right], t\in[0, 1]\Big\}$ where $T_0$ is a given real constant, then we have

In fact, note that in this case we have $S(0) = S(T) = \mathbb{R}$ and $M_{21}(0) = 0, M_{21}(T) = T_0$, then it is just the consequence of (A.2).

Acknowledgements

The authors thank the anonymous referees for the excellent and substantial work to evaluate the manuscript as well as for the several suggestions proposed, leading to the improvement of the manuscript to its final form.

A special thank Prof. Xijun Hu for providing excellent working conditions in Shandong University and for the useful and deep discussions about this project.

Li Wu is supported by NSFC (Nos.12171281, 12071255) and "The Fundamental Research Funds of Shandong University". Ran Yang is partially supported by NSFC(N.12001098) and Doctoral research start-up fund of East China University of Technology (N.DHBK2019204).

Conflict of interest

The authors declare there is no conflicts of interest.